A note on chaotic unimodal maps and applications

Based on the word-lift technique of symbolic dynamics of one-dimensional unimodal maps, we investigate the relation between chaotic kneading sequences and linear maximum-length shift-register sequences. Theoretical and numerical evidence that the set of the maximum-length shift-register sequences is a subset of the set of the universal sequence of one-dimensional chaotic unimodal maps is given. By stabilizing unstable periodic orbits on superstable periodic orbits, we also develop techniques to control the generation of long binary sequences.     

Image encryption based on new Beta chaotic maps

In this paper, we created new chaotic maps based on Beta function. The use of these maps is to generate chaotic sequences. Those sequences were used in the encryption scheme. The proposed process is divided into three stages: Permutation, Diffusion and Substitution. The generation of different pseudo random sequences was carried out to shuffle the position of the image pixels and to confuse the relationship between the encrypted the original image, so that significantly increasing the resistance to attacks. The acquired results of the different types of analysis indicate that the proposed method has high sensitivity and security compared to previous schemes.     

Multifractal Analysis of Asymptotically Additive Sequences

For saturated maps, we effect a complete multifractal analysis of the dimension spectra obtained from asymptotically additive sequences of continuous functions. This includes, for example, the class of maps with the specification property. We consider also the more general cases of ratios of sequences and of multidimensional spectra in which a single sequence is replaced by a vector of sequences. In addition, we establish a conditional variational principle for the topological pressure of a continuous function on the level sets of an asymptotically additive sequence (again in the former general setting). Finally, we apply our results to the dimension spectra of an average conformal repeller. In particular, we obtain almost automatically a conditional variational principle for the Hausdorff dimension of the level sets obtained from an asymptotically additive sequence.     

On the role of semisimple subalgebras in the deformation theory of lie algebras and their homomorphisms general theory and applications

Formal deformations of Lie algebras are determined by sequences of bilinear alternating maps, and those of their homomorphisms by sequences of linear maps. The question of the existence, in any equivalence class of formal deformations of Lie algebras and of their homomorphisms, of elements determined by well-behaved sequences is investigated in this paper. A satisfactory affirmative answer is given provided the Lie algebra to be deformed has a semisimple subalgebra different from {0}. The meaning of this result in the geometric approach to deformation theory is pointed out. Applications to the problem of coupling the Poincaré group and an internal symmetry group in a nontrivial way and to the study of deformations of irreducible finite-dimensional representations of $\text{E}$(3) are given.     

Entropic Bounds on Semiclassical Measures for Quantized One-Dimensional Maps

Quantum ergodicity asserts that almost all infinite sequences of eigenstates of quantized ergodic Hamiltonian systems are equidistributed in phase space. This, however, does not prohibit existence of exceptional sequences which might converge to different (non-Liouville) classical invariant measures. It has been recently shown by N. Anantharaman and S. Nonnenmacher in [20,21] (with H. Koch) that for Anosov geodesic flows the metric entropy of any semiclassical measure μ must satisfy a certain bound. This remarkable result seems to be optimal for manifolds of constant negative curvature, but not in the general case, where it might become even trivial if the (negative) curvature of the Riemannian manifold varies a lot. It has been conjectured by the same authors, that in fact, a stronger bound (valid in the general case) should hold. In the present work we consider such entropic bounds using the model of quantized piecewise linear one-dimensional maps. For a certain class of maps with non-uniform expansion rates we prove the Anantharaman-Nonnenmacher conjecture. Furthermore, for these maps we are able to construct some explicit sequences of eigenstates which saturate the bound. This demonstrates that the conjectured bound is actually optimal in that case.     

Quadratic maps without asymptotic measure

An interval map is said to have an asymptotic measure if the time averages of the iterates of Lebesgue measure converge weakly. We construct quadratic maps which have no asymptotic measure. We also find examples of quadratic maps which have an asymptotic measure with very unexpected properties, e.g. a map with the point mass on an unstable fix point as asymptotic measure. The key to our construction is a new characterization of kneading sequences.     

Symbolic Dynamics for the Tél Map

In terms of contracting and expanding foliations the two-dimensional Tél map is decomposed into two coupled one-dimensional maps. Symbolic sequences are assigned to the two classes of foliations, and their ordering is discussed. The pruning front is constructed. A necessary and sufficient condition for admissible sequences is proposed.     

Jacobian elliptic Chebyshev rational maps

We introduce the Jacobian elliptic Chebyshev rational maps with the the equidistributivity property (or briefly EDP) and the semi-group property like the Chebyshev polynomial maps. Simple methods are also discussed for generating sequences of i.i.d. p-ary random variables based on the Jacobian elliptic Chebyshev rational maps. Furthermore, we give conjectures on correlational properties of a real-valued trajectory generated by the Jacobian elliptic Chebyshev rational maps.     

Various Auto-Correlation Functions of m-Bit Random Numbers Generated from Chaotic Binary Sequences

This paper discusses the auto-correlation functions of m-bit random numbers obtained from m chaotic binary sequences generated by one-dimensional nonlinear maps. First, we provide the theoretical auto-correlation function of an m-bit sequence obtained by m binary sequences that are assumed to be uncorrelated to each other. The auto-correlation function is expressed by a simple form using the auto-correlation functions of the binary sequences. This implies that the auto-correlation properties of the m-bit sequences can be easily controlled by the auto-correlation functions of the original binary sequences. In numerical experiments using a computer, we generated m-bit random sequences using some chaotic binary sequences with prescribed auto-correlations generated by one-dimensional chaotic maps. The numerical experiments show that the numerical auto-correlation values are almost equal to the corresponding theoretical ones, and we can generate m-bit sequences with a variety of auto-correlation properties. Furthermore, we also show that the distributions of the generated m-bit sequences are uniform if all of the original binary sequences are balanced (i.e., the probability of 1 (or 0) is equal to 1/2) and independent of one another.     

Recent advances in rice genome and chromosome structure research by fluorescence in situ hybridization (FISH)

Fluorescence in situ hybridization (FISH) is an effective method for the physical mapping of genes and repetitive DNA sequences on chromosomes. Physical mapping of unique nucleotide sequences on specific rice chromosome regions was performed using a combination of chromosome identification and highly sensitive FISH. Increases in the detection sensitivity of smaller DNA sequences and improvements in spatial resolution have ushered in a new phase in FISH technology. Thus, it is now possible to perform in situ hybridization on somatic chromosomes, pachytene chromosomes, and even on extended DNA fibers (EDFs). Pachytene-FISH allows the integration of genetic linkage maps and quantitative chromosome maps. Visualization methods using FISH can reveal the spatial organization of the centromere, heterochromatin/euchromatin, and the terminal structures of rice chromosomes. Furthermore, EDF-FISH and the DNA combing technique can resolve a spatial distance of 1 kb between adjacent DNA sequences, and the detection of even a 300-bp target is now feasible. The copy numbers of various repetitive sequences and the sizes of various DNA molecules were quantitatively measured using the molecular combing technique. This review describes the significance of these advances in molecular cytology in rice and discusses future applications in plant studies using visualization techniques.     

Symbolic approach to intermittency

We propose the use of the MSS universal sequences in numerical and experimental studies of type I intermittency. Pattern repetitions in these symbolic sequences can be easily measured and provide a reliable and unambiguous way of testing intermitency scaling laws and the predictions of one-dimensional maps. The limitations of the conventional acceptance gate method in intermittency are discussed. Universality of MSS sequences makes our approach, in contrast with the previous one, independent of particular features of the map.     

Efficiency of purging sequences in the long-range heteronuclear shift correlation experiment

The efficiencies of a number of pulse sequences designed to remove directly bonded C-H correlations from long-range C-H shift correlation maps are evaluated. A two-step J filter sequence is shown to give good suppression in 1 D experiments. Its incorporation into the long-range C-H shift correlation experiment with a BIRD sequence at the center of the refocusing period gives the BIRDTRAP sequence, which is shown to yield 2D maps with a few very weak direct correlations and no artifacts. BIRDTRAP has a sensitivity higher than that of FLOCK.     

Opacity of an automaton. Application to the inhomogeneous Ising chain

An automaton maps infinite sequences onto infinite sequences. We define the opacity as the distance between output sequences and input sequences. A transparent automaton hardly disturbs the input sequence. An opaque automaton erases some of the information contained in the input sequence. We apply these ideas to the study of the inhomogeneous Ising chain governed by the Hamiltonian      

Efficient Entropic Security with Joint Compression and Encryption Approach Based on Compressed Sensing with Multiple Chaotic Systems

This paper puts forward a new algorithm that utilizes compressed sensing and two chaotic systems to complete image compression and encryption concurrently. First, the hash function was utilized to obtain the initial parameters of two chaotic maps, which were the 2D-SLIM and 2D-SCLMS maps, respectively. Second, a sparse coefficient matrix was transformed from the plain image through discrete wavelet transform. In addition, one of the chaotic sequences created by 2D-SCLMS system performed pixel transformation on the sparse coefficient matrix. The other chaotic sequences created by 2D-SLIM were utilized to generate a measurement matrix and perform compressed sensing operations. Subsequently, the matrix rotation was combined with row scrambling and column scrambling, respectively. Finally, the bit-cycle operation and the matrix double XOR were implemented to acquire the ciphertext image. Simulation experiment analysis showed that the compressed encryption scheme has advantages in compression performance, key space, and sensitivity, and is resistant to statistical attacks, violent attacks, and noise attacks.     

Universality of k·3n and k·4n bifurcations in area-preserving maps

We study period-trebling and period-quadrupling bifurcations in two-dimensional reversible area-preserving maps. Our numerical results show that there are unique universal limiting behaviors in each of the period-trebling and period-quadrupling sequences.     

Stabilizing long-period orbits via symbolic dynamics in simple limiter controllers

We present an efficient approach to determine the control parameter of simple limiter controllers by using symbolic dynamics of one-dimensional unimodal maps. By applying addition- and subtraction-symbol rules for generating an admissible periodic sequence, we deal with the smallest base problem of the digital tent map. The proposed solution is useful for minimizing the configuration of digital circuit designs for a given target sequence. With the use of the limiter controller, we show that one-dimensional unimodal maps may be robustly employed to generate the maximum-length shift-register sequences. For an arbitrary long Sarkovskii sequence, the control parameters are analytically given.     

Binary tree approach to scaling in unimodal maps

Ge, Rusjan, and Zweifel introduced a binary tree which represents all the periodic windows in the chaotic regime of iterated one-dimensional unimodal maps. We consider the scaling behavior in a modified tree which takes into account the self-similarity of the window structure. A nonuniversal geometric convergence of the associated superstable parameter values towards a Misiurewicz point is observed for almost all binary sequences with periodic tails. For these sequences the window period grows arithmetically down the binary tree. There are an infinite number of exceptional sequences, however, for which the growth of the window period is faster. Numerical studies with a quadratic maximum suggest more rapid than geometric scaling of the superstable parameter values for such sequences.     

混沌伪随机序列的复杂度的稳定性研究

增强统计复杂度能反映混沌伪随机序列的随机本质,在此基础上提出了k错增强统计复杂度的定义,用来衡量混沌伪随机序列复杂度的稳定性,并证明了其两个基本特性.以Logistic,Henon,Cubic,Chebyshev和Tent映射产生的混沌伪随机序列为例,说明了该方法的应用.仿真结果表明,该方法能区分不同混沌伪随机序列的稳定性,是一种衡量混沌序列稳定性的有效方法. 关键词： 稳定性 k错增强统计复杂度')" href="#">k错增强统计复杂度 混沌 伪随机序列     

Complex mixed-mode periodic and chaotic oscillations in a simple three-variable model of nonlinear system

A detailed study of a generic model exhibiting new type of mixed-mode oscillations is presented. Period doubling and various period adding sequences of bifurcations are observed. New type of a family of 1D (one-dimensional) return maps is found. The maps are discontinuous at three points and consist of four branches. They are not invertible. The model describes in a qualitative way mixed-mode oscillations with two types of small amplitude oscillations at local maxima and local minima of large amplitude oscillations, which have been observed recently in the Belousov-Zhabotinsky system. (c) 2000 American Institute of Physics.